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Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle X} can equivalently be defined as an equivalence relation on X {\displaystyle X} , together with a partial order on the set of equivalence class.
Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, := < is a non-strict partial order. Thus, if ≤ {\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b if a ≤ b and a ...
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations . The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves ...
A chain with 1 element has length 0, one with 2 elements has length 1, etc. Linear. See total order. Linear extension. A linear extension of a partial order is an extension that is a linear order, or total order. Locale. A locale is a complete Heyting algebra. Locales are also called frames and appear in Stone duality and pointless topology.
Furthermore, a natural preorder of elements of the underlying set of a topology is given by the so-called specialization order, that is actually a partial order if the topology is T 0. Conversely, in order theory, one often makes use of topological results.
It follows that a space X is T 0 if and only if the specialization preorder ≤ on X is a partial order. There are numerous partial orders on a finite set. Each defines a unique T 0 topology. Similarly, a space is R 0 if and only if the specialization preorder is an equivalence relation.
Firstly, the order type of the set of natural numbers is ω. Any other model of Peano arithmetic, that is any non-standard model, starts with a segment isomorphic to ω but then adds extra numbers. For example, any countable such model has order type ω + (ω* + ω) ⋅ η. Secondly, consider the set V of even ordinals less than ω ⋅ 2 + 7:
Preorder (Quasiorder) Partial order Total preorder Total order Prewellordering Well-quasi-ordering Well-ordering Lattice Join-semilattice Meet-semilattice Strict partial order Strict weak order Strict total order Symmetric: Antisymmetric: Connected: Well-founded: Has joins: Has meets: Reflexive: Irreflexive: Asymmetric